The Inverse Surfaces of Tangent Developables with Respect to S {C}(R)

Home , Tangent developable

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2 Basic notions of inverse surfaces

∗ Let c ∈ E3 and r ∈ R+. We denote that E3 = E3 −{c} . Then, an inversion of E3 with the center c ∈ E3 and the radius r is the map ∗ ∗ Φ[c, r]: E3 −→ E3 given by r2 Φ[c, r] (p)= c + (p − c) . (2.1) kp − ck2 Deﬁnition 2.1. ([7]) Let Φ [c, r] be an inversion with the center c and the ∗ radius r Then, the tangent map of Φ at p ∈ E3 is the map En ∗ En ∗ Φ∗p : Tp ( ) −→ TΦ(p) ( ) given by 2 2 r vp 2r h(p − c) , vpi Φ∗p (vp)= − (p − c) , kp − ck2 kp − ck4

3 ∗ where vp ∈ T p E .

∗ Now, let us assume that X : U ⊂ E2 −→ E3 is the parametrization of a surface. The inverse surface of X with respect to Φ [c, r] is the surface given by

Y = Φ [c, r] ◦ X, (2.2) Throughout this paper, we assume that Φ is an inversion of E3 with the ∗ center c and the radius r, X is a patch in E3 and Y is inverse patch of X with respect to Φ. Let IX, IIX and KX, HX be the ﬁrst and second fundamental forms and the curvatures (Gauss and mean) of X, and let IY, IIY and KY, HY be these of Y, respectively. From [1], we have

2 IY ◦ Φ∗ = λ IX, (2.3)

IIY ◦ Φ∗ = −λIIX − 2δIX, (2.4)

1 4 −1 4 2 KY = KX + λ ηHX + η , (2.5) λ2 r2 r4

2 1 2η H = − H − , (2.6) Y λ X r2 2 2 r 2r hUX,(X−c)i λ 2 δ 4 η U , X c where = kX−ck , = kX−ck and = h X ( − )i.

3 The tangent developable surface

Let γ : I ⊂ R −→ E3 be a curve with arc-length s and {T,N,B} be Frenet frame along γ. Denote by κ and τ the curvature and the torsion of the curve γ, respectively. Then we have Frenet formulas T ′ (s) = κ (s) N (s) , N ′ (s) = −κ (s) T (s)+ τ (s) B (s) , B′ (s) = −τ (s) N (s) . The tangent developable of γ is a ruled surface parametrized by M (s,u)= γ (s)+ uT (s) , (3.1) where T is unit tangent vector field of γ. As it is known, the coefficients of the first and second fundamental forms of the surface M (s,u) have following 2 EM =1+(uκ) , FM = GM =1, (3.2) and eM = −sgn (uκ) (uκτ) , fM = gM =0. (3.3) The normal vector field of the surface M (s,u) is given by

UM (s,u)= −sgn (uκ) B (s) . (3.4) Next the curvatures (mean and Gaussian) and the matrix of shape operator of this surface are respectively as follows −sgn (uκ) τ H = , and K = 0 (3.5) M 2uκ M and −sgn (uκ) τ 1 1 SM = . (3.6) 2uκ 0 0 Finally, the Christoﬀel symbols of the surface M (s,u) are given by uκ + κ Γ1 = s , 11 M uκ 2 −κ 1 + (uκ) − uκs Γ2 = , (3.7) 11 M uκ 1 Γ1 = − Γ2 = , 12 M 12 M u 1 2 Γ22 M = Γ22 M =0.

3 4 The inverse surface of the tangent developable

We show that N is the inverse surface of the tangent developable surface M with respect to the inversion Φ. Thus the inverse surface N has following parametrization r2 N = c + (M − c) . (4.1) kM − ck2 Hence, if we take into account the equalities (2.3) and (2.4) , then the coef- ﬁcients of the ﬁrst and second fundamental forms of the inverse surface N by the help of these of the surface M are given by

2 2 2 EN = λ 1 + (uκ) , FN = GN = λ , (4.2) 2 lN = sgn (uκ) λuκτ − 2δ 1 + (uκ) , mN = nN = −2δ, (4.3) where EN, FN, GN and lN, mN, nN are the coeﬃcients of the ﬁrst and second fundamental forms of the inverse surface N, respectively. Morever,the Gauss and mean curvatures of the inverse surface N by the help of these of the surface M are respectively, using by (2.5) and (2.6) , 4 τ η K η sgn uκ , N = 2 − ( ) + 2 (4.4) r 2λuκ r τ 2η H = sgn (uκ) − , (4.5) N 2λuκ r2 where KN and HN are the Gauss and the mean curvatures of the inverse surface N, respectively.

Theorem 4.1. Let N be the inverse surface of the tangent developable surface M with respect to the inversion Φ. Denote by SN the matrix of the shape operator of the inverse surface N, then SN is given by the help of that of M as follows

τ 2η τ sgn (uκ) λuκ − r2 sgn (vκ) λvκ SN = 2η . (4.6) 0 − r2

Proof. Let SM be the matrix of the shape operator of surface M. By using the equalities (2.3) and (2.4) , we can write 2 S ◦ Φ = −λ−1S − ηI , (4.7) N ∗ M r2 2

2 r I λ 2 η U , M c . . where 2 is identity, = kM−ck and = h M ( − )i Hence from (3 6) and (4.7) , we obtain that the equality (4.6) is satisﬁed.

4 i Theorem 4.2. Let Γjk be the Christoﬀel symbols of the inverse surface N N. The Christoﬀel symbols of the inverse surface N by the help of these of the surface M are given by

2 2 2 ∂λ 2 ∂λ uκ + κ (uκ) − 1 ∂s + (uκ) +1 ∂u Γ1 = s + , 11 N uκ 2 2 2λ (uκ) 2 2 2 2 2 ∂λ 2 ∂λ −κ 1 + (uκ) − uκs (uκ) +1 ∂s + (uκ) +1 ∂u Γ2 = + , 11 N uκ 2 2 2λ (uκ) 2 2 2 ∂λ ∂λ 1 (uκ) +1 ∂u − ∂s Γ1 = + , 12 N u 2 2 2λ (uκ) 2 2 2 ∂λ ∂λ 1 (uκ) +1 ∂s − ∂u Γ2 = − + , 12 N u 2 2 2λ (uκ) 2 2 ∂λ ∂λ ∂u − ∂s 1 , Γ22 N = 2 2 2λ (uκ) 2 2 2 ∂λ ∂λ (uκ) − 1 ∂u + ∂s 2 . Γ22 N = 2 2 2λ (uκ)

Proof. Considering the equality (2.3) , for i, j, k =1, we can write

2 2 2 ∂λ ∂λ EMGM − 2FM + FMEM 1 1 ∂s ∂u Γ11 = Γ11 + , N M 2λ2 [E G − F 2 ] M M M 2 r 1 λ 2 where = kM−ck and Γ11 M is the Christoﬀel symbol of the tangent developable surface. Thus, from the equalities (3.2) and (3.7) , we obtain

2 2 2 ∂λ 2 ∂λ uκ + κ (uκ) − 1 ∂s + (uκ) +1 ∂u Γ1 = s + 11 N uκ 2 2 2λ (uκ) Others are found in similar way.

Theorem 4.3 Let N be the inverse surface of the tangent developable surface M with respect to the inversion Φ. Then the inverse surface N is a ﬂat surface if and only if either the normal lines to the surface M or the tangent planes of the surface M pass through the center of inversion.

5 Proof. Let us assume that the inverse surface N is flat, then from (4.4) , we can write 4 τ η η sgn uκ , 2 − ( ) + 2 =0 (4.9) r 2λuκ r where either η = hUM, (M − c)i =0, (4.10) or τ η sgn (uκ) = . (4.11) 2λuκ r2 If the equality (4.10) is satisfied, then the tangent planes of the surface M pass through the center of inversion. If the equality (4.11) holds, then it follows τ U = sgn (uκ) (M − c) . M 2uκ Namely, the normal lines to the surface M pass through the center of inversion. The proof of sufficient condition is obvious.

Theorem 4.4 Let N be the inverse surface of the tangent developable surface M with respect to Sc (r) . The inverse surface N is minimal if and only if the normal lines to the surface M pass through the center of inversion

Proof. The proof is same with that of Theorem 4.1.

Applications.

Fig 1.

6 Fig 2.

Fig 1: The helicoid given by (u cos v,u sin v, 2v) .

Fig 2: The inverse surface of the helicoid with respect to unit sphere given by u u 2v cos v, sin v, . u2 +4v2 u2 +4v2 u2 +4v2

References

[1] A. Gray: Modern diﬀerential geometry of curves and surfaces with mathe- matica. CRC Press LLC, 1998. [2] D. A. Brannan, M. F. Esplen, J. J. Gray : Geometry, Cambridge Universtiy Press, Cambridge, 1999. [3] D. E. Blair: Inversion theory and conformal mapping. American Mathe- matical Society, 2000. [4] E. Ozyilmaz, Y. Yayli: On the closed space-like developable ruled surface, Hadronic J. 23 (4) (2000) 439–456. [5] H. S. M. Coexeter: Inversive Geometry, Educational Studies in Mathemat- ics, 3 (1971), 310-321. [6] F. Beardona, D. Mindap: Sphere-Preserving Maps in Inversive Geometry. Proceedings of the American Mathematical Society, 130(4) (2001), 987-998. [7] PM. do Carmo : Riemann Geometry. Birkhauser, Boston, 1992. [8] P. Alegre, K. Arslan, A. Carriazo, C. Murathan and G. Ozt¨urk:¨ Some Spe- cial Types of Developable Ruled Surface, Hacettepe Journal of Mathematics and Statistics, 39 (3) (2010), 319 – 325.

7 [9] S. Izumiya, H. Katsumi and T. Yamasaki: The rectifying developable and the spherical Darboux image of a space curve. Geometry and topology of caustics-Caustics ’98- Banach Center Publications, 50 (1999), 137-149. [10] S. Izumiya and N. Takeuchi: Special curves and ruled surfaces, Applicable Mathematics in the Golden Age (ed., J.C. Misra), Narosa Publishing House, New Delhi, (2003) 305-338. [11] S. Izumiya, N. Takeuchi: New Special Curves and Developable Surfaces, Turk J Math 28 (2004), 153-163.